Recently in [1]}, a generalization of topological transitivity, called \(\mathcal {F}\) -Transitivity, where \(\mathcal {F}\)  is a family, was investigated. Recall that a non-empty
collection of subsets of \(\mathbb {N}\)  is a Furstenberg family or just a family if it satisfies the following property:
\(\emptyset \notin \mathcal {F}\ \textrm {and if}\ A \in \mathcal {F}\ \textrm {and}\ A \subset B,\ \textrm {then}\ B \in \mathcal {F}.\) 
